Simple example

Example taken from.^[3]

Consider the vector field ${\displaystyle ({\dot {x}},{\dot {y}})=(-y-x^{3},x^{5})}$ in the plane. The function ${\displaystyle V(x,y)=x^{6}+3y^{2}}$ satisfies ${\displaystyle {\dot {V}}=-6x^{8}}$, and is radially unbounded, showing that the origin is globally asymptotically stable.

Pendulum with friction

This section will apply the invariance principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation^[4]

${\displaystyle ml{\ddot {\theta }}=-mg\sin \theta -kl{\dot {\theta }}}$

where ${\displaystyle \theta }$ is the angle the pendulum makes with the vertical normal, ${\displaystyle m}$ is the mass of the pendulum, ${\displaystyle l}$ is the length of the pendulum, ${\displaystyle k}$ is the friction coefficient, and g is acceleration due to gravity.

This, in turn, can be written as the system of equations

${\displaystyle {\dot {x}}_{1}=x_{2}}$

${\displaystyle {\dot {x}}_{2}=-{\frac {g}{l}}\sin x_{1}-{\frac {k}{m}}x_{2}}$

Using the invariance principle, it can be shown that all trajectories that begin in a ball of certain size around the origin ${\displaystyle x_{1}=x_{2}=0}$ asymptotically converge to the origin. We define ${\displaystyle V(x_{1},x_{2})}$ as

${\displaystyle V(x_{1},x_{2})={\frac {g}{l}}(1-\cos x_{1})+{\frac {1}{2}}x_{2}^{2}}$

This ${\displaystyle V(x_{1},x_{2})}$ is simply the scaled energy of the system.^[4] Clearly, ${\displaystyle V(x_{1},x_{2})}$ is positive definite in an open ball of radius ${\displaystyle \pi }$ around the origin. Computing the derivative,

${\displaystyle {\dot {V}}(x_{1},x_{2})={\frac {g}{l}}\sin x_{1}{\dot {x}}_{1}+x_{2}{\dot {x}}_{2}=-{\frac {k}{m}}x_{2}^{2}}$

Observe that ${\displaystyle V(0)={\dot {V}}(0)=0}$. If it were true that ${\displaystyle {\dot {V}}<0}$, we could conclude that every trajectory approaches the origin by Lyapunov's second theorem. Unfortunately, ${\displaystyle {\dot {V}}\leq 0}$ and ${\displaystyle {\dot {V}}}$ is only negative semidefinite since ${\displaystyle x_{1}}$ can be non-zero when ${\displaystyle {\dot {V}}=0}$. However, the set

${\displaystyle S=\{(x_{1},x_{2})|{\dot {V}}(x_{1},x_{2})=0\}}$

which is simply the set

${\displaystyle S=\{(x_{1},x_{2})|x_{2}=0\}}$

does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time ${\displaystyle t}$, ${\displaystyle x_{2}(t)=0}$, then because ${\displaystyle x_{1}}$ must be less than ${\displaystyle \pi }$ away from the origin, ${\displaystyle \sin x_{1}\neq 0}$ and ${\displaystyle {\dot {x}}_{2}(t)\neq 0}$. As a result, the trajectory will not stay in the set ${\displaystyle S}$.

All the conditions of the local version of the invariance principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as ${\displaystyle t\rightarrow \infty }$.^[5]